For (a), we note that \(Z^{(1),s}_{\min }\)  is disjoint from \(C^{(1)}(X)\) , as stable points have zero dimensional stabiliser and we assumed \(Z^{(1),s}_{\min } \ne Z^{(1),ss}_{\min }\) . Thus \(B^{(1)}(X):=p_{(1)}^{-1}(C^{[1]}(X))\)  is disjoint from \(X^{(1),s}_{\min }\) . We then conclude (a) as \(X^{s}_{\min } \subset X^{(1),s}_{\min }\)  by Remark REF . Note that \(X^{s}_{\min } \ne \emptyset \) , by Assumption (REF ) of (QiS).
For (b), we have seen that \(Z^{s}_{\min }\)  is disjoint from the centre of the blow-up in (a) and since \(Z^{s}_{\min } \subset Z^{[i],s}_{\min }\)  (resp. \(Z^{s}_{\min } \subset Z^{(i),s}_{\min }\) ), we see that \(Z^{[i]}_{\min }\)  (resp. \(Z_{\min }\) ) is not contained in the centre of the blow-up. Consequently, we deduce (b).
For (c), to deduce REF  holds for \(\widehat{X}_{(1)}\) , we use Lemma REF  for the groups \(U^{[i]}\)  and \(U^{[i-1,i]}\) , together with the fact that the dimensions of these stabilisers on \(X^{[i],ss}_{\min }\)  are constant. We can directly deduce that Assumptions (REF ) and (REF ) of (QiS) hold for \(\widehat{X}_{(1)}\)  as the strict transform of \(Z_{\min }^{[i]}\)  (resp. \(Z_{\min }^{(i)}\) ) is \(Z_{\min }^{[i]}(\widehat{X})\)  (resp. \(Z_{\min }^{(i)}(\widehat{X})\) ) by (c).
Finally, non-degeneracy (REF ) of (QiS) holds for \(\widehat{X}_{(1)}\)  as we can identify \(X^{s}_{\min }\)  with a non-empty open set of \(\widehat{X}_{(1),\min }^s\)  by (a).
For (d), we can use (b) to identify \(Z_{\min }^{(1)}(\widehat{X})\)  with the strict transform of \(Z_{\min }^{(1)}\) . Then the result follows exactly as in [1]}.
